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We present results for the equilibrium statistics and dynamic evolution ofmoderately large ($n = {\mathcal{O}}(10^2 - 10^3)$) numbers of interactingpoint vortices on the unit sphere under the constraint of zero mean angularmomentum. We consider a binary gas consisting of equal numbers of vortices withpositive and negative circulations. When the circulations are chosen to beproportional to $1/\sqrt{n}$, the energy probability distribution function,$p(E)$, converges rapidly with $n$ to a function that has a single maximum,corresponding to a maximum in entropy. Ensemble-averaged wavenumber spectra ofthe nonsingular velocity field induced by the vortices exhibit the expected$k^{-1}$ behavior at small scales for all energies. The spectra at the largestscales vary continuously with the inverse temperature $\beta$ of the system andshow a transition from positively sloped to negatively sloped as $\beta$becomes negative. The dynamics are ergodic and, regardless of the initialconfiguration of the vortices, statistical measures simply relax towardsmicrocanonical ensemble measures at all observed energies. As such, thedirection of any cascade process measured by the evolution of the kineticenergy spectrum depends only on the relative differences between the initialspectrum and the ensemble mean spectrum at that energy; not on the energy, ortemperature, of the system.

Ergodicity and spectral cascades in point vortex flows on the sphere